Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2205 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C3 |
| prime radical | C3×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ G | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(3,11)(4,14)(5,16)(6,10)(7,12)(8,15)(9,13)(17,18,19);(1,3,6,13)(2,7,10,14)(4,15,12,5)(8,9,16,11);(1,4,6,12)(2,8,10,16)(3,5,13,15)(7,11,14,9)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5496);